A finite volume scheme preserving extremum principle for convection-diffusion equations on polygonal meshes

2017 ◽  
Vol 84 (10) ◽  
pp. 616-632 ◽  
Author(s):  
Qi Zhang ◽  
Zhiqiang Sheng ◽  
Guangwei Yuan
Keyword(s):  
Finite Volume ◽  
Extremum Principle ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Polygonal Meshes ◽  
Convection Diffusion ◽  
Volume Scheme

scholarly journals Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 407-432 ◽  
Author(s):  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Stella Krell
Keyword(s):  
Finite Volume ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Time Behavior ◽  
Convection Diffusion ◽  
Volume Scheme ◽  
Existence Of Positive Solutions ◽  
Long Time ◽  
Nonlinear Form ◽  
Compactness Arguments

AbstractWe propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy/energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

scholarly journals Nonlinear Finite Volume Scheme Preserving Positivity for 2D Convection-Diffusion Equations on Polygonal Meshes

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Bin Lan ◽  
Jianqiang Dong
Keyword(s):  
Finite Volume ◽  
Upper Bound ◽  
Finite Volume Scheme ◽  
Polygonal Meshes ◽  
Present Scheme ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Volume Scheme ◽  
Arbitrary Convex ◽  
Steady Convection

In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.

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